Q: Can I hold my rooster upside down by the feet to calm him down?
A: No. That's a terrible idea.
Birds' respiratory systems are completely different ours, so holding your rooster upside down can cause him real problems. For example, a bird's lungs are right next to the spine and upper ribcage, so it is hard for a bird that is upside down to breathe, since the weight of all his organs will be pressing on his lungs and some of his air sacs. Normally, he will breathe via changes in pressure in the air sacs. So... when the pressure changes "unnaturally," it can make it difficult for him to compensate. Chicken's lungs can't expand like ours do, either.
Being hung upside down is pretty uncomfortable for humans, but we breathe by using a muscle called the diaphragm, which is a dome shaped muscle between the chest and the abdomen. Birds don't have a diaphragm---they have air sacs, and breathe by changes in pressure in their air sacs. Some of the air sacs even extend into their bones (pneumatic bones), and the sacs act as a bellows to ventilate the lungs. They include cervical sacs, interclavicular sac, humeral sacs (these are parts of the interclavicular sac that extend into the bones of the humeris), thoracic sacs (anterior and posterior), abdominal sacs and so forth.
The reason holding a rooster upside down to "calm" him works is that he can't breathe properly. Birds have been known to die this way. If you have an aggressive rooster, we recommend you try the method listed in the related questions below for addressing that aggression.